At the beginning, prioritize mastery over breadth. I find that it's very necessary to have at least one champion you can become amazing with if you don't want to just get brutalized over and over. Get to Mastery 7 and winning your lane/position (not necessarily game) 80% of the time or better with at least one champion - this probably will take a few months if you're doing it right.
When you graduate to draft/ranked play on a regular basis, that expands to two, because you'll need to survive at least one ban on your position. And that's two per position you plan to play - you can't always get the one you want - so ideally you should be able to play at least two positions well and have four really good champions before playing draft/ranked regularly.
If you get really serious, learn to play every position well - so ten champions, two in each of all five roles.
After that, breadth is the goal. Really top level players can probably whomp the average forum denizen here with a few dozen champions. But even they still have certain champs they just utterly thrive with, so always keep at least a few champs that you just master to the limit and stay current on!